Triangle Inequality Theorem

Unit 5: Relationships Within Triangles · ~15–30 min

Not just any three lengths make a triangle

Pick three lengths at random and try to build a triangle out of them — sometimes it works, sometimes the two shorter sides simply can't stretch far enough to meet. The Triangle Inequality Theorem tells you exactly when it's possible.

• Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

This has to hold for all three pairings of sides — but in practice, you only ever need to check the pairing of the two shortest sides against the longest one. If that pairing works, the other two automatically will too.

Guided practice

• 1.Can lengths 4, 4, and 9 form a triangle?
  Show answer

  No — 4 + 4 = 8, which is not greater than 9, so the inequality fails.
• 2.Can lengths 6, 8, and 10 form a triangle?
  Show answer

  Yes — 6 + 8 = 14 > 10, so the inequality holds.
• 3.Two sides of a triangle measure 10 and 10. Find the range of the third side, x.
  Show answer

  x < 10 + 10 = 20 and x > 10 − 10 = 0, so 0 < x < 20.
• 4.Two sides measure 12 and 5. Which of these could be the third side: 6, 9, or 18?
  Show answer

  The range is 12 − 5 = 7 < x < 12 + 5 = 17. Of the three choices, only 9 falls inside that range, so only 9 works.
• 5.Why does checking only the two shortest sides against the longest side guarantee the whole triangle inequality holds?
  Show answer

  If the two shortest sides already sum to more than the longest side, then pairing either of them with the longest side will sum to even more than the remaining (shorter) side, so those pairings are automatically satisfied.